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VLSI Arithmetic Adders & Multipliers. Prof. Vojin G. Oklobdzija University of California http://www.ece.ucdavis.edu/acsel. Digital Computer Arithmetic belongs to Computer Architecture, however, it is also an aspect of logic design.

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## VLSI Arithmetic Adders & Multipliers

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**VLSI ArithmeticAdders & Multipliers**Prof. Vojin G. Oklobdzija University of California http://www.ece.ucdavis.edu/acsel**Digital Computer Arithmetic belongs to Computer**Architecture, however, it is also an aspect of logic design. The objective of Computer Arithmetic is to develop appropriate algorithms that are utilizing available hardware in the most efficient way. Ultimately, speed, power and chip area are the most often used measures, making a strong link between the algorithms and technology of implementation. Introduction Computer Arithmetic**Addition**Multiplication Multiply-Add Division Evaluation of Functions Multi-Media Basic Operations Computer Arithmetic**Addition of Binary Numbers**Full Adder. The full adder is the fundamental building block of most arithmetic circuits: The sum and carry outputs are described as: ai bi Full Adder Cout Cin si Computer Arithmetic**Inputs**Outputs ci ai bi si ci+1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Addition of Binary Numbers Propagate Generate Propagate Generate Computer Arithmetic**Full-Adder Implementation**Full Adder operations is defined by equations: Carry-Propagate: and Carry-Generate gi One-bit adder could be implemented as shown Computer Arithmetic**High-Speed Addition**One-bit adder could be implemented more efficiently because MUX is faster Computer Arithmetic**The Ripple-Carry Adder**Computer Arithmetic**The Ripple-Carry Adder**From Rabaey Computer Arithmetic**Inversion Property**From Rabaey Computer Arithmetic**Minimize Critical Path by Reducing Inverting Stages**From Rabaey Computer Arithmetic**Ripple Carry Adder**Carry-Chain of an RCA implemented using multiplexer from the standard cell library: Critical Path Oklobdzija, ISCAS’88 Computer Arithmetic**Manchester Carry-Chain Realization of the Carry Path**• Simple and very popular scheme for implementation of carry signal path Computer Arithmetic**Original Design**T. Kilburn, D. B. G. Edwards, D. Aspinall, "Parallel Addition in Digital Computers: A New Fast "Carry" Circuit", Proceedings of IEE, Vol. 106, pt. B, p. 464, September 1959. Computer Arithmetic**Manchester Carry Chain (CMOS)**• Implement P with pass-transistors • Implement G with pull-up, kill (delete) with pull-down • Use dynamic logic to reduce the complexity and speed up Kilburn, et al, IEE Proc, 1959. Computer Arithmetic**Pass-Transistor Realization in DPL**Computer Arithmetic**Carry-Skip Adder**MacSorley, Proc IRE 1/61 Lehman, Burla, IRE Trans on Comp, 12/61 Computer Arithmetic**Carry-Skip Adder**Bypass From Rabaey Computer Arithmetic**Carry-Skip Adder:N-bits, k-bits/group, r=N/k groups**Computer Arithmetic**Carry-Skip Adder**k Computer Arithmetic**Variable Block Adder(Oklobdzija, Barnes: IBM 1985)**Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) 6 5 5 4 4 3 3 D=9 1 1 Any-point-to-any-point delay = 9 D as compared to 12 D for CSKA Computer Arithmetic**Carry-chain block size determination for a 32-bit Variable**Block Adder(Oklobdzija, Barnes: IBM 1985) Computer Arithmetic**Delay Calculation for Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Delay model: Computer Arithmetic**Variable Block Adder(Oklobdzija, Barnes: IBM 1985)**Variable Group Length Oklobdzija, Barnes, Arith’85 Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Variable Block Lengths • No closed form solution for delay • It is a dynamic programming problem Computer Arithmetic**Delay Comparison: Variable Block Adder(Oklobdzija, Barnes:**IBM 1985) Computer Arithmetic**Delay Comparison: Variable Block Adder**VBA CLA VBA- Multi-Level Computer Arithmetic**VLSI ArithmeticLecture 4**Prof. Vojin G. Oklobdzija University of California http://www.ece.ucdavis.edu/acsel**Review**Lecture 3**Variable Block Adder(Oklobdzija, Barnes: IBM 1985)**Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) 6 5 5 4 4 3 3 D=9 1 1 Any-point-to-any-point delay = 9 D as compared to 12 D for CSKA Computer Arithmetic**Carry-chain block size determination for a 32-bit Variable**Block Adder(Oklobdzija, Barnes: IBM 1985) Computer Arithmetic**Delay Calculation for Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Delay model: Computer Arithmetic**Variable Block Adder(Oklobdzija, Barnes: IBM 1985)**Variable Group Length Oklobdzija, Barnes, Arith’85 Computer Arithmetic**Carry-chain of a 32-bit Variable Block Adder(Oklobdzija,**Barnes: IBM 1985) Variable Block Lengths • No closed form solution for delay • It is a dynamic programming problem Computer Arithmetic**Delay Comparison: Variable Block Adder(Oklobdzija, Barnes:**IBM 1985) Computer Arithmetic**Delay Comparison: Variable Block Adder**Square Root Dependency VBA Log Dependency CLA VBA- Multi-Level Computer Arithmetic**Circuit Issues**• Adder speed can not be estimated based on: • logic gates in the critical path • number of transistors in the path • logic levels in the path • Estimating Adders speed is much more complex and many of the “fast” schemes may be misleading you. Computer Arithmetic**Fan-Out Dependency**Computer Arithmetic**Fan-In Dependency**This looks like “Logical Effort” (1985) Computer Arithmetic**Delay Comparison: Variable Block Adder(Oklobdzija, Barnes:**IBM 1985) Computer Arithmetic**Carry-Lookahead Adder(Weinberger and Smith, 1958)**ARITH-13: Presenting Achievement Award to Arnold Weinberger of IBM (who invented CLA adder in 1958) Ref: A. Weinberger and J. L. Smith, “A Logic for High-Speed Addition”, National Bureau of Standards, Circ. 591, p.3-12, 1958. Computer Arithmetic**CLA Definitions: One-bit adder**Computer Arithmetic**CLA Definitions: 4-bit Adder**Computer Arithmetic**Carry-Lookahead Adder: 4-bits**Gj Pj Computer Arithmetic

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